Although a long time ago, when i applied to do medicine I was told not to bother applying to UCL or Royal Free as no one from my school had ever got in. ( a top 10 independent) I went ahead and applied to Royal free and the interview experience was horrific. Exceedingly prejudiced against private education.
I got places at UCL and Royal Free (but it may have been even longer ago
) - the interviews were fine and hardly mentioned my school. At UCL there was a very wide range of med students from comprehensives to well known indies - didn't seem to make much difference. There may well have been a certain political bias at UCL but they didn't seem to let it influence the who they made offers to.